5/11/017 PHYS485 Materials Physics Dr. Gregory W. Clar Manchester University LET S GO ON A (TEK)ADVENTURE! WHAT? TRIP TO A MAKER S SPACE IN FORT WAYNE WHEN? THURSDAY, MAY 11 TH @ 5PM WHERE? TEKVENTURE 1
5/11/017 Effect of an electric field Shift of Fermi sphere Center now displaced from origin by Current flow represented by electrons with unmatched values of ; fits with Ohm s Law original Fermi surface y E J nevd ne E / m E where τ is the ave. time between e- collisions x Successes? Free electron gas model explains electrical conductivity reasonably well for some materials (e.g., low carrier concentrations); close to classical result Explains classical Hall effect Does not completely explain heat capacity, thermal conductivity, thermoelectric effect, magnetic properties, photoelectric effect, etc.
5/11/017 Born-von Karman BC! (i.e., wavefunctions at ends are equal) Source: Kings College ( x) Ce Qx De Qx Qx Qx 3 ( x) ( Ce De ) e i ( a b ) Boundary Conditions: 1 ( 0) (0) 1 ( x) Ae ikx Be ikx d 1 ( 0) d (0) dx dx 1 ( a) 3 ( a) ( b) e i ( a b ) d dx 1 ( a) d 3 ( a) d ( b) i ( a b ) dx dx e 3
5/11/017 Delta function limit: ( Q b / K ) sin Ka cos Ka cos a LHS Forbidden values in green! Ka E E 4
5/11/017 Fermi Surface Copper Group Velocity In 1-D: In 3-D: v d d v 1 E 1 de d E where e ˆ i i i velocity is perpendicular to Fermi surface 5
5/11/017 Equation of Motion Can relate external forces to wavevector via: F ext F ext dp dt d dt Can relate acceleration to: a 1 d E d F ext So that we can define the effective mass: m * 1 d E d 1 Particle behaves as if has mass m * in external force Effective Mass m * 1 d E d 1 Can be + (E vs. concave up), - (E vs. concave down) or infinite (inflection points). E v = (1/ħ) de/d (dv/d) -1 = (d E/d )-1 dv/d = (1/ħ) d E/d 6
5/11/017 Effective Mass Frequently negative near FBZ boundary An e - with m * < 0 can be thought of as a hole with m p* > 0 In 3-D: Effective Mass Tensor F ext m * a where 1 m * ij 1 E i j Acceleration, in general, not parallel to external force Semiconductors 7
5/11/017 Semiconductors Si & Ge: the mainstays Silicon and Germanium crystallize in the diamond structure: two interpenetrating FCC. Lattice parameters: Si is 0.543 nm Ge is 0.566 nm 8
5/11/017 Intrinsic Semiconductors E g small enough that some e- excited to CB at RT: e - /h + pairs Intrinsic semiconductors 9
5/11/017 Semiconductors Si band structure: indirect gap E g (ev) InSn 0.18 InAs 0.36 Ge 0.67 Si 1.11 GaAs 1.43 SiC.3 ZnS 3.6 C (dia) 5.5 Density of Occupied/Unoccupied States Build on what we now: simple model Thin Kronig-Penney-lie Consider points in nd (=VB) and 3 rd (=CB) bands, at ~ 0 VB is hole-lie: m* < 0 CB is e - -lie: m* > 0 Free-electron-lie model using effective masses E m * E E C E g E V 10
5/11/017 Density of States Must have n = p for intrinsic (pure) semiconductors This implies that the Fermi level (chem pot l) is 1 1 ( " EF ") ( EC EV ) BT ln( NV / NC ) EX: For Si: c.p. to metals? n ~?? n ~ 10 8 m -3 At T = 300 K, N C ~ N V ~ 3 x 10 5 m -3 >>> n i = p i ~10 16 m -3 At T = 373 K, n i = p i ~10 18 m -3 (100-fold increase!) 1 ( " EF ") ( EC EV ) (as we assumed early on) Doping To increase the number of carriers we can dope the semiconductor, using appropriate impurities that have more/fewer e - than the intrinsic SC. Impurities that contribute electrons to the conduction band are called donors. Impurities that contribute holes to the valence band are called acceptors. If a semiconductor contains both donors and acceptors it is called compensated, potentiallly yielding no free carriers. 11
5/11/017 Consider the Periodic Table Conductivity of semiconductors can be modified by addition of impurities. The process of adding impurities is called doping and the impurities are called dopants. Two types: n-type and p-type Al Si must be very pure due to extreme sensitivity to impurities (99.9999% pure). 1
5/11/017 n-type and p-type n-type semiconductors Donor atoms majority carriers: electrons minority carriers: holes 13
5/11/017 p-type semiconductors Acceptor atoms majority carriers: holes minority carriers: electrons p-n junction Consists of two semiconducting regions of opposite type with a common interface Many technological apps: rectification, isolation, V-dependent capacitor solar cells, photodiodes, light emitting diodes (LEDs), laser diodes Basic element of bipolar junction transistors (BJTs) and field effect transistors (FETs) 14
5/11/017 A first loo at the p-n junction Imagine joining together the two different types of SC (perfectly, so XTAL structure varies smoothly at border!) Interesting things happen at the junction! A first loo at the p-n junction There are drift and diffusion currents in the materials A built-in voltage difference between the conductions bands of the two materials occurs as the Fermi energies align. diffusion & annihilation! creation of depletion zone 15
5/11/017 p-n junction Formation of depletion layer & CPD Diffusion of majority carriers + recombination creates depletion layer (depletion of majority carries) contact potential difference http://www.acsu.buffalo.edu/~wie/applet/pnformation/pnformation.html http://www.yena.com/freecontent/attachment.action?quic=14r&att=93 16
5/11/017 The carriers Carrier particle and concentration majority carriers minority carriers n-type e -, n n h +, p n p-type h +, p p e -, n p p p n n n p p n 17
5/11/017 Applications Rectification V regulation Photovoltaics LEDs Light emitting diode A forward biased diode can act as an LED As e - enter the CB in the p region, the recombination process emits radiation (similarly with h + in the VB) The radiation emitted depends on the band gap, which can be tuned (in direct gap SCs) to specific values of hν 18